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  "body": "\n## Definition\n\nLet $k$ be a field and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $k$.\nLet $\\operatorname{mod} A$ be the category of finitely generated left $A$-[[modules|0NZ8JA25WK]].\n\n1. A morphism $f \\colon M \\to N$ between [[indecomposable|E2P29GCD9T]] modules is *irreducible* if it is neither a split monomorphism nor a split epimorphism, and the following holds.\nIn every factorization $f = g h$, either $h$ is a split monomorphism or $g$ is a split epimorphism.\n2. For $M$ in $\\operatorname{mod} A$ with minimal [[projective|BFFA03GE3V]] presentation $P_1 \\xrightarrow{p} P_0 \\to M \\to 0$, the *transpose* $\\operatorname{Tr} M$ is the cokernel of $\\Hom_A(p, A) \\colon \\Hom_A(P_0, A) \\to \\Hom_A(P_1, A)$, a right $A$-module.\n3. The *Auslander–Reiten translation* is $\\tau M = D \\operatorname{Tr} M$, where $D = \\Hom_k(\\cdot, k)$.\n\n*Auslander–Reiten theory* is the study of $\\operatorname{mod} A$ through irreducible morphisms, the translation $\\tau$, [[almost split sequences|M5D95FN7G3]] and the [[Auslander–Reiten quiver|125Z033VCQ]].\n\n## Remarks\n\nThe basic existence theorem: if $M$ is indecomposable and not projective, there is an almost split sequence $0 \\to \\tau M \\to E \\to M \\to 0$.\nDually, for $N$ indecomposable and not injective there is an almost split sequence starting at $N$, and it ends at $\\tau^{-1} N = \\operatorname{Tr} D N$.\nExample: for the path algebra of $1 \\to 2$, with simple modules $S_1, S_2$ and the projective cover $P_1$ of $S_1$, one has $\\tau S_1 = S_2$.\nThe almost split sequence is $0 \\to S_2 \\to P_1 \\to S_1 \\to 0$.\nThe translation $\\tau$ sends projective modules to $0$.\nThe theory works more generally for Artin algebras over a commutative Artinian ring $R$, with $D = \\Hom_R(\\cdot, E)$ for the injective envelope $E$ of $R/\\operatorname{rad} R$.\n",
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